LMFDB - Newform orbit 5929.2.a.x

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5929,2,Mod(1,5929)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5929, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5929.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5929 = 7^{2} \cdot 11^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5929.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.3433033584$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 3x - 1$ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{2} + (\beta_{2} - \beta_1 + 1) q^{3} + \beta_{2} q^{4} + (\beta_{2} + 2) q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{6} + (\beta_1 - 1) q^{8} + (2 \beta_{2} - 3 \beta_1) q^{9} + ( - 3 \beta_1 - 1) q^{10}+ \cdots + ( - \beta_{2} + \beta_1 + 15) q^{97}+O(q^{100})$ q - b1 * q^2 + (b2 - b1 + 1) * q^3 + b2 * q^4 + (b2 + 2) * q^5 + (b2 - 2b1 + 1) q^6 + (b1 - 1) * q^8 + (2b2 - 3b1) * q^9 + (-3b1 - 1) q^10 + q^12 + (-b2 - b1 - 1) * q^13 + (2b2 - 2b1 + 3) * q^15 + (-3b2 + b1 - 2) q^16 + (4b2 - b1 + 1) q^17 + (3b2 - 2b1 + 4) * q^18 + (2b2 - b1 - 3) q^19 + (b2 + b1 + 2) * q^20 + (2b2 - 5b1) * q^23 + (-2b2 + 3b1 - 2) * q^24 + (3b2 + b1 + 1) q^25 + (b2 + 2b1 + 3) q^26 + (-3b1 + 2) q^27 + (3b2 - 4b1 + 1) * q^29 + (2b2 - 5b1 + 2) * q^30 + (-b2 + 3) * q^31 + (-b2 + 3b1 + 3) q^32 + (b2 - 5b1 - 2) q^34 + (-2b2 - b1 + 1) q^36 + (-2b2 + 4b1) * q^37 + (b2 + b1) * q^38 + (-b1 - 1) * q^39 + (-b2 + 3b1 - 1) q^40 + (3b2 - b1 - 3) q^41 + (b2 + b1) * q^43 + (2b2 - 7b1 + 1) * q^45 + (5b2 - 2b1 + 8) * q^46 + (-6b2 + 3b1 - 1) * q^47 + (-3b2 + 4b1 - 6) * q^48 + (-b2 - 4b1 - 5) q^50 + (2b2 - 3b1 + 6) * q^51 + (-2b1 - 3) q^52 + (-3b2 - 3b1 + 3) * q^53 + (3b2 - 2b1 + 6) * q^54 + (-2b2 + b1) q^57 + (4b2 - 4b1 + 5) * q^58 + (5b2 + b1) q^59 + (b2 + 2) * q^60 + (-4b2 + 6b1 - 4) * q^61 + (-2b1 + 1) q^62 + (3b2 - 4b1 - 1) * q^64 + (-2b2 - 4b1 - 5) * q^65 + (2b2 - 2b1) * q^67 + (-3b2 + 3b1 + 7) * q^68 + (5b2 - 10b1 + 7) * q^69 + (2b2 + 5b1 - 3) * q^71 + (-5b2 + 5b1 - 4) * q^72 + (b2 - b1 - 2) * q^73 + (-4b2 + 2b1 - 6) * q^74 + (b1 + 3) * q^75 + (-5b2 + b1 + 3) q^76 + (b2 + b1 + 2) * q^78 + (5b2 + 2b1 + 1) * q^79 + (-5b2 - 9) q^80 + (-b2 + b1 + 5) * q^81 + (b2 - 1) * q^82 + (-2b2 + 5b1 + 5) * q^83 + (5b2 + b1 + 9) q^85 + (-b2 - b1 - 3) * q^86 + (5b2 - 9b1 + 8) * q^87 + (b2 - 5) * q^89 + (7b2 - 3b1 + 12) * q^90 + (-2b2 - 3b1 - 1) * q^92 + (3b2 - 3b1 + 2) * q^93 + (-3b2 + 7b1) * q^94 + (-b2 - b1 - 3) * q^95 + (3b1 - 1) q^96 + (-b2 + b1 + 15) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{3} + 6 q^{5} + 3 q^{6} - 3 q^{8} - 3 q^{10} + 3 q^{12} - 3 q^{13} + 9 q^{15} - 6 q^{16} + 3 q^{17} + 12 q^{18} - 9 q^{19} + 6 q^{20} - 6 q^{24} + 3 q^{25} + 9 q^{26} + 6 q^{27} + 3 q^{29} + 6 q^{30}+ \cdots + 45 q^{97}+O(q^{100})$ 3 * q + 3 * q^3 + 6 * q^5 + 3 * q^6 - 3 * q^8 - 3 * q^10 + 3 * q^12 - 3 * q^13 + 9 * q^15 - 6 * q^16 + 3 * q^17 + 12 * q^18 - 9 * q^19 + 6 * q^20 - 6 * q^24 + 3 * q^25 + 9 * q^26 + 6 * q^27 + 3 * q^29 + 6 * q^30 + 9 * q^31 + 9 * q^32 - 6 * q^34 + 3 * q^36 - 3 * q^39 - 3 * q^40 - 9 * q^41 + 3 * q^45 + 24 * q^46 - 3 * q^47 - 18 * q^48 - 15 * q^50 + 18 * q^51 - 9 * q^52 + 9 * q^53 + 18 * q^54 + 15 * q^58 + 6 * q^60 - 12 * q^61 + 3 * q^62 - 3 * q^64 - 15 * q^65 + 21 * q^68 + 21 * q^69 - 9 * q^71 - 12 * q^72 - 6 * q^73 - 18 * q^74 + 9 * q^75 + 9 * q^76 + 6 * q^78 + 3 * q^79 - 27 * q^80 + 15 * q^81 - 3 * q^82 + 15 * q^83 + 27 * q^85 - 9 * q^86 + 24 * q^87 - 15 * q^89 + 36 * q^90 - 3 * q^92 + 6 * q^93 - 9 * q^95 - 3 * q^96 + 45 * q^97

Basis of coefficient ring in terms of $\nu = \zeta_{18} + \zeta_{18}^{-1}$ :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 2$ v^2 - 2

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 2$ b2 + 2

Display $\beta_i$ in terms of $\nu^j$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

1.87939
−0.347296
−1.53209

−1.87939

0.652704

1.53209

3.53209

−1.22668

0.879385

−2.57398

−6.63816

1.2

0.347296

−0.532089

−1.87939

0.120615

−0.184793

−1.34730

−2.71688

0.0418891

1.3

1.53209

2.87939

0.347296

2.34730

4.41147

−2.53209

5.29086

3.59627

Atkin-Lehner signs

$p$	Sign
$7$	$+1$
$11$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5929.2.a.x		3
7.b	odd	2	1		5929.2.a.u		3
7.c	even	3	2		847.2.e.c		6
11.b	odd	2	1		539.2.a.j		3
33.d	even	2	1		4851.2.a.bj		3
44.c	even	2	1		8624.2.a.ch		3
77.b	even	2	1		539.2.a.g		3
77.h	odd	6	2		77.2.e.a	✓	6
77.i	even	6	2		539.2.e.m		6
77.m	even	15	8		847.2.n.f		24
77.o	odd	30	8		847.2.n.g		24
231.h	odd	2	1		4851.2.a.bk		3
231.l	even	6	2		693.2.i.h		6
308.g	odd	2	1		8624.2.a.co		3
308.n	even	6	2		1232.2.q.m		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.2.e.a	✓	6	77.h	odd	6	2
539.2.a.g		3	77.b	even	2	1
539.2.a.j		3	11.b	odd	2	1
539.2.e.m		6	77.i	even	6	2
693.2.i.h		6	231.l	even	6	2
847.2.e.c		6	7.c	even	3	2
847.2.n.f		24	77.m	even	15	8
847.2.n.g		24	77.o	odd	30	8
1232.2.q.m		6	308.n	even	6	2
4851.2.a.bj		3	33.d	even	2	1
4851.2.a.bk		3	231.h	odd	2	1
5929.2.a.u		3	7.b	odd	2	1
5929.2.a.x		3	1.a	even	1	1	trivial
8624.2.a.ch		3	44.c	even	2	1
8624.2.a.co		3	308.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5929))$ :

T_{2}^{3} - 3T_{2} + 1

T_{3}^{3} - 3T_{3}^{2} + 1

T_{5}^{3} - 6T_{5}^{2} + 9T_{5} - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3} - 3T + 1$ T^3 - 3*T + 1
$3$	$T^{3} - 3T^{2} + 1$ T^3 - 3*T^2 + 1
$5$	$T^{3} - 6 T^{2} + \cdots - 1$ T^3 - 6T^2 + 9T - 1
$7$	$T^{3}$ T^3
$11$	$T^{3}$ T^3
$13$	$T^{3} + 3 T^{2} + \cdots + 1$ T^3 + 3T^2 - 6T + 1
$17$	$T^{3} - 3 T^{2} + \cdots + 127$ T^3 - 3T^2 - 36T + 127
$19$	$T^{3} + 9 T^{2} + \cdots + 9$ T^3 + 9T^2 + 18T + 9
$23$	$T^{3} - 57T - 107$ T^3 - 57*T - 107
$29$	$T^{3} - 3 T^{2} + \cdots - 51$ T^3 - 3T^2 - 36T - 51
$31$	$T^{3} - 9 T^{2} + \cdots - 19$ T^3 - 9T^2 + 24T - 19
$37$	$T^{3} - 36T + 72$ T^3 - 36*T + 72
$41$	$T^{3} + 9 T^{2} + \cdots + 1$ T^3 + 9T^2 + 6T + 1
$43$	$T^{3} - 9T - 9$ T^3 - 9*T - 9
$47$	$T^{3} + 3 T^{2} + \cdots - 323$ T^3 + 3T^2 - 78T - 323
$53$	$T^{3} - 9 T^{2} + \cdots + 459$ T^3 - 9T^2 - 54T + 459
$59$	$T^{3} - 93T + 19$ T^3 - 93*T + 19
$61$	$T^{3} + 12 T^{2} + \cdots + 24$ T^3 + 12T^2 - 36T + 24
$67$	$T^{3} - 12T - 8$ T^3 - 12*T - 8
$71$	$T^{3} + 9 T^{2} + \cdots - 801$ T^3 + 9T^2 - 90T - 801
$73$	$T^{3} + 6 T^{2} + \cdots + 1$ T^3 + 6T^2 + 9T + 1
$79$	$T^{3} - 3 T^{2} + \cdots - 37$ T^3 - 3T^2 - 114T - 37
$83$	$T^{3} - 15 T^{2} + \cdots + 267$ T^3 - 15T^2 + 18T + 267
$89$	$T^{3} + 15 T^{2} + \cdots + 111$ T^3 + 15T^2 + 72T + 111
$97$	$T^{3} - 45 T^{2} + \cdots - 3329$ T^3 - 45T^2 + 672T - 3329
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